(i) Consider the integral equation

$$\phi(x)=-\lambda \int_{a}^{b} K(x, t) \phi(t) d t+g(x)$$

for $\phi$ in the interval $a \leq x \leq b$, where $\lambda$ is a real parameter and $g(x)$ is given. Describe the method of successive approximations for solving ( $\dagger$ ).

Suppose that

$$|K(x, t)| \leq M, \quad \forall x, t \in[a, b]$$

By using the Cauchy-Schwarz inequality, or otherwise, show that the successive-approximation series for $\phi(x)$ converges absolutely provided

$$|\lambda|<\frac{1}{M(b-a)} .$$

(ii) The real function $\psi(x)$ satisfies the differential equation

$$-\psi^{\prime \prime}(x)+\lambda \psi(x)=h(x), \quad 0<x<1$$

where $h(x)$ is a given smooth function on $[0,1]$, subject to the boundary conditions

$$\psi^{\prime}(0)=\psi(0), \quad \psi(1)=0 .$$

By integrating $(\star)$, or otherwise, show that $\psi(x)$ obeys

$$\psi(0)=\frac{1}{2} \int_{0}^{1}(1-t) h(t) d t-\frac{1}{2} \lambda \int_{0}^{1}(1-t) \psi(t) d t$$

Hence, or otherwise, deduce that $\psi(x)$ obeys an equation of the form ( $\dagger$ ), with

$$\begin{gathered} K(x, t)= \begin{cases}\frac{1}{2}(1-x)(1+t), & 0 \leq t \leq x \leq 1 \\ \frac{1}{2}(1+x)(1-t), & 0 \leq x \leq t \leq 1\end{cases} \\ \text { and } g(x)=\int_{0}^{1} K(x, t) h(t) d t \end{gathered}$$

Deduce that the series solution for $\psi(x)$ converges provided $|\lambda|<2$.